Non-Homogeneous Hydrodynamic Systems and Quasi-Stäckel Hamiltonians

Abstract

Abstract. In this paper we present a novel construction of non-homogeneous hydrodynamic equations from what we call quasi-Stäckel systems, that is non-commutatively integrable systems constructed from appropriate maximally superintegrable Stäckel systems. We describe the relations between Poisson algebras generated by quasi-Stäckel Hamiltonians and the corresponding Lie algebras of vector fields of non-homogeneous hydrodynamic systems. We also apply Stäckel transform to obtain new non-homogeneous equations of considered type.

Systematic construction of nonautonomous Hamiltonian equations of Painlevé type. I. Frobenius integrability

“…With this choice of 𝑣 𝑖𝑘 , the Hamiltonians  𝑟 in (12) span a (𝑚-dependent) Lie algebra 𝔤 = span{ 𝑟 ∈ 𝐶 ∞ ()∶ 𝑟 = 1, … , 𝑛} with the following commutation relations: 22 { 1 ,  𝑟 } = 0, 𝑟 = 2, … , 𝑛, and…”

“…In Ref. 22, we introduced and investigated the so-called geodesic quasi-Stäckel Hamiltonians. These Hamiltonians constitute a noncommuting finite-dimensional Lie algebra with respect to the Poisson bracket, and thus, evolution equations they generate are not Frobenius integrable.…”

“…Further, both conditions are met if (a sufficient condition) the functions 𝑣 𝑖𝑘 are chosen as The formulas (13) constitute a definition of the set of functions 𝑣 𝑖𝑘 for various 𝑚 (thus 𝑣 𝑖𝑘 depend on 𝑚). With this choice of 𝑣 𝑖𝑘 , the components of Killing vector fields 𝐽 𝑟 are given explicitly by Ref 22.…”

Non-autonomous Hénon-Heiles system from Painlevé class

“…In Hamiltonian formalism on M = T * Q, with such system one can relates n geodesic Hamiltonians E 1 , ..., E n in involution and n Hamiltonian vector fields X 1 , ..., X n that commute. Next, extend geodesic Hamiltonians E i → h i = E i + W i , i = 2, ..., n by linear in momenta terms, generated by Killing vectors of g in such a way that h i constitute a Lie algebra [14]. Then, add separable potentials h i → h i = E i + W i + V i and prove for which ones there exists a non-autonomous deformation h i → H i (t 1 , ..., t n ) satisfying the Frobenius condition (5).…”

“…W 2 = −p 1 is generated by the Killing vector Z = (−1, 0) T of the Euclidean metric in R 2 . Second, add to both Hamiltonians the lower nontrivial positive separable potentials (14) with coefficients depending on evolution parameters, i.e. c 3 (t 1 , t 2 )V (3) + c 2 (t 1 , t 2 )V (2) .…”